Robust self-testing of multipartite GHZ-state measurements in quantum networks

The rapid development of quantum communication in recent years creates an exigent requirement for devising certification methods to guarantee correctness of quantum information tasks. To rule out any potential attacks by malicious third party, such certification methods must be device-independent. As the first device-independent tool, the Bell nonlocality has been extensively studied in recent decades [1]. It has brought great breakthroughs in quantum physics. Recently, as the strongest form of device-independent certification, self-testing has been developed, which is also based on Bell nonlocality. Such certification method can characterize the target objects (quantum states, measurements) fully, only up to local isometries, in a device-independent manner.

Self-testing, acting as a device-independent certification method, has attracted lots of attention since the pioneer works of Mayers and Yao [2]. It can be used to certify entangled pure states and measurements [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Up to now, a wide range of entangled quantum states are proved to be self-testable, such as the elegant results for all pure bipartite entangled states [23], three-qubit W states [24], and graph states [25]. It has also been shown that all pure multipartite GHZ states and Dicke states can be self-tested [26]. Recently, the self-testing method for quantum channels has also been developed [27]. Moreover, there have been many applications about self-testing, such as quantum key distribution [28], randomness expansion [29], detection for entanglement [30], certification of genuinely entangled subspaces [31, 32], coarse-grained self-testing of a many-body singlet [33], as well as verification of quantum computations [34, 35].

In this work, we will focus on self-testing entangled measurements in quantum networks. Self-testing entangled quantum measurements is of great potential to develop practical quantum networks, which has been preliminarily studied [36, 37]. For a star-network as shown in Fig.  1, where $N$ observers share entangled states with central node, respectively. A self-testable entangled measurement can guarantee the success of quantum information tasks, which are based on distributing entangled states between remote parties in such a network. Meanwhile, the entanglement between observers and central node can also be certified. In Ref.  [36], the authors presented a self-testing method for the Bell-state measurement (BSM) and three-qubit GHZ-state measurement (GSM). Furthermore, a more robust self-testing scheme for BSM has also been proposed in Ref.  [37]. However, there have not been a detailed characterization for self-testing multipartite ($N>3$) entangled measurements directly.

By generalizing the idea of Ref.  [36], we present herewith a self-testing method for $N$-qubit tilted GSM, whose eigenstates are partially entangled $N$-qubit GHZ states (tilted GHZ states). That is to say, one of the eigenstates of $N$-qubit tilted GSM can be written as $|GHZ^{0}_{\theta}\rangle=\cos\theta|0\rangle^{\otimes N}+\sin\theta|1\rangle^{\otimes N},\ \theta\in(0,\pi/4].$ In the $N+1$-partite star-network shown in Fig.  1, after performing measurement in central node (Roy), the remaining particles shared with Alice $1$, Alice $2$,$\dots$, and Alice $N$ will be projected into the eigenstates of Roy’s measurement. If the particular correlations of remaining particles are observed by local measurements on Alice $1$, Alice $2$,$\dots$, and Alice $N$, the measurement performed by Roy is equivalent to an $N$-qubit tilted GSM, up to a local isometry. A local isometry is a linear local operation on quantum states that preserves inner products [38]. Thus, the first step to self-test tilted GSM is self-testing all its measurement eigenstates, and the problem of self-testing tilted GSM can be converted to the problem of self-testing states. Motivated by the method of self-testing multipartite entangled states in Ref.  [26], we have developed further a general method for self-testing multipartite tilted GSM in a star-network, and the method is operational from an experimental point of view. We also show that one can self-test more entangled measurements by our developed method straightforwardly.

The paper is organized as following. In Sec.  2, we provide a preparation review of tilted CHSH scenario which constitute important ingredient of our self-testing method. In Sec.  3, self-testing method of multipartite tilted GSM is presented. In Sec.  4, a noise-robust self-testing scheme of three-qubit GSM is presented, with the help of semidefinite program (SDP) method. Finally, we conclude our results and make a discussion on potential future works in Sec.  5.

To self-test tilted multipartite GSM, the tilted CHSH inequality is necessary [39]. Let us consider a task: Alice and Bob share a two-qubit state and they want to know whether the shared state is partially entangled or not. They perform local measurements (dichotomic observables) respectively. The tilted CHSH inequality is given by

where the maximal value of violation is $\sqrt{8+2\alpha^{2}},\alpha\in[0,2)$, $A_{i}$ and $B_{i}$ being observables with outcomes $\{-1,+1\}$ measured locally by Alice and Bob. Here, we omit the notation “$\otimes$” between systems $A$ and $B$ and write $A_{0}\otimes I$ as $A_{0}$ for short. After performing local measurements, if Alice and Bob obtain the maximal violation of tilted CHSH inequality, the state shared by them is a certain partially entangled two-qubit state (tilted Bell state). For detailed case, the four tilted Bell states $|Bell^{b}_{\theta}\rangle,b=0,1,2,3$ are given by

where $c_{\theta}=\cos\theta,s_{\theta}=\sin\theta,\theta\in(0,\frac{\pi}{4}]$. Let $\mu$ satisfy $\tan\mu=\sin 2\theta$ and $\sigma_{Z},\sigma_{X}$ be Pauli matrices. If one fixes the measurement settings of Alice and Bob as $A_{0}=\sigma_{Z},\ A_{1}=\sigma_{X},\ B_{0}=\cos\mu\sigma_{Z}+\sin\mu\sigma_{X},\ B_{1}=\cos\mu\sigma_{Z}-\sin\mu\sigma_{X}$, the output statistics obtained by these measurements will maximally violate some tilted CHSH inequalities. The maximal violation is $CHSH_{b}^{\alpha}=\langle Bell^{b}_{\theta}|W_{b}^{\alpha}|Bell^{b}_{\theta}\rangle=\sqrt{8+2\alpha^{2}}$ with $\alpha=2\cos 2\theta/\sqrt{1+\sin^{2}2\theta}$, where

Here the $W_{b}^{\alpha}$ is Bell operator acting on the Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ of Alice and Bob. It is easy to show that the eigenvalue $\sqrt{8+2\alpha^{2}}$ of the Bell operator $W_{b}^{\alpha}$ is nondegenerate with associated eigenvector $|Bell^{b}_{\theta}\rangle$. Hence, if the maximal violation of $CHSH_{b}^{\alpha}$ is $\sqrt{8+2\alpha^{2}}$, the shared state will be $|Bell^{b}_{\theta}\rangle$. One can discriminate the four tilted Bell states by the maximally violations of four tilted Bell inequality with fix measurement settings. Furthermore, other tilted Bell states that are local-unitary (constructed by $\sigma_{Z},\sigma_{X}$) equivalent to the above four tilted Bell states can also be discriminated. For example, the state $|\Phi\rangle=c_{\theta}|00\rangle-s_{\theta}|11\rangle=\sigma_{Z_{A}}|Bell_{\theta}^{0}\rangle$. It can maximally violate tilted CHSH inequality with $CHSH_{\theta}=\langle\sigma_{Z_{A}}W_{0}^{\alpha}\sigma_{Z_{A}}^{{\dagger}}\rangle=\alpha\langle A_{0}\rangle+\langle A_{0}B_{0}+A_{0}B_{1}-A_{1}B_{0}+A_{1}B_{1}\rangle$ and fixed measurements given above.

In the entanglement swapping scenario [40] shown in Fig.  2, let Charlie perform tilted BSM whose measurement eigenstates are tilted Bell states with outcomes $b$. Then, the remaining state will be projected into one of the four tilted Bell states $|Bell^{b}_{\theta}\rangle$ conditioned on the outcomes $b$. Conversely, if one finds that Alice and Bob share tilted Bell states $|Bell^{b}_{\theta}\rangle$ for $b\in\{0,1,2,3\}$, the performed measurement of Charlie is a tilted BSM. Motivated by this idea, we will develop a procedure for preforming self-testing of tilted multipartite GSM.

As shown in Ref.  [41], any completely positive and trace preserving (CPTP) map can be implemented, by tracing out degrees of freedom that does not involve effective information after applying a local isometry. Therefore, one can adopt the approach presented in [15, 36, 38] to present the definition for self-testing multipartite measurements via simulation: denote an ideal $d$-outcome measurement for Roy acting on $\mathcal{H}_{R_{1}^{\prime}}\otimes\mathcal{H}_{R_{2}^{\prime}}\otimes\dots\otimes\mathcal{H}_{R_{N}^{\prime}}$ as $\mathcal{P}^{\prime}=\{P_{R_{1}^{\prime}R_{2}^{\prime}\dots R_{N}^{\prime}}^{\prime r}\}_{r=1}^{d}$, and a real measurement acting on $\mathcal{H}_{R_{1}}\otimes\mathcal{H}_{R_{2}}\otimes\dots\otimes\mathcal{H}_{R_{N}}$ as $\mathcal{P}=\{P_{R_{1}R_{2}\dots R_{N}}^{r}\}_{r=1}^{d}$. If there exist completely positive and unital maps $\Lambda_{R_{j}}:\ \mathcal{L}(\mathcal{H}_{R_{j}})\to\mathcal{L}(\mathcal{H}_{A_{j}^{\prime}})$, for $j\in\{1,2,\dots,N\}$, such that

for all $r$, we say $\mathcal{P}$ is capable of simulating $\mathcal{P}^{\prime}$. In the above definition, we adopt the assumption that the different physical sources are independent in a quantum network. The construction of a quantum network as shown in Fig.  1 guarantees the well-defined $N$-partition for Roy’s measurement device, i.e., $\mathcal{H}_{R}=\mathcal{H}_{R_{1}}\otimes\mathcal{H}_{R_{2}}\otimes\dots\otimes\mathcal{H}_{R_{N}}$.

The idea of our self-testing method relies on the task of entanglement swapping as shown in Fig.  1. There are $N$ initially uncorrelated parties Alice $1$ ($A_{1}$), Alice $2$ ($A_{2}$),$\dots$, Alice $N$ ($A_{N}$). They are independently entangled with an additional party, Roy. Specifically, the $A_{i}$ and Roy share a Bell state $|Bell^{0}_{\pi/4}\rangle_{A_{i}R_{i}}\in\mathcal{H}_{A_{i}}\otimes\mathcal{H}_{R_{i}},i\in\{1,2,\dots,N\}$. To distribute entanglement between $A_{1},A_{2},\dots,A_{N}$ in such a quantum network, Roy performs the tilted GSM and obtains outcomes $r\in\{k_{1}k_{2}\dots k_{N}\}$ with $k_{1},k_{2},\cdots,k_{N}\in\{0,1\}$. For simplicity, we denote the outcomes as $r\in\{0,1,\dots,2^{N}-1\}$. Then, the states shared by $A_{1},A_{2},\dots,A_{N}$ are projected to one of the $2^{N}$ tilted GHZ states $|GHZ^{r}_{\theta}\rangle$ based on the outcome $r$. The tilted GHZ states are measurement eigenstates of tilted GSM given by

where $\bar{k}_{i}=1-k_{i}$ and $k_{i}\in\{0,1\},\ i\in\{1,\dots,N\}$. The tilted GSM can be denoted as $GSM_{\theta}=\{GHZ^{r}_{\theta}\}_{r=0}^{2^{N}-1}$, with $GHZ^{r}_{\theta}=|GHZ^{r}_{\theta}\rangle\langle GHZ^{r}_{\theta}|$. If $A_{1},A_{2},\dots,A_{N}$ obtain the outcomes $r$ of Roy, they can apply a special local unitary operation on their qubits, so that they share a special tilted GHZ state. With the above operations, we have implemented distribution of entanglement between $N$ remote parties.

The self-testing procedure is similar to the task of entanglement swapping, without assumptions on the dimensions, initial states and operators. From now on, let us adopt labels on the same letter to make a distinction between two Hilbert spaces, e.g., $Q$ and $Q^{\prime}$. Specially, the $Q^{\prime}$ is in a two-dimensional Hilbert space and the dimension of $Q$ is unknown. Let us start with presenting a self-testing method for $N$-partite tilted GHZ states given in Ref.  [26].

The junk state in the Lemma 1 can be any state and can be removed by tracing out the $A_{1}A_{2}\dots A_{N}$ space. It should be noted that this self-testing method is also suitable for a general $\rho$ [26]. Without loss of generality, let the $N$-partite state be a pure state. Here, the $Z_{N}$ and $X_{N}$ act on $|\psi\rangle$ in the same way as $(A_{0,N}+A_{1,N})/(2\cos\mu)$ and $(A_{0,N}-A_{1,N})/(2\sin\mu)$, respectively [38]. For details, the ideal measurements achieving these correlations in the Lemma 1 are: $A_{0,i}^{\prime}=\sigma_{Z}$, $A_{1,i}^{\prime}=\sigma_{X}$, for $i=1,\dots,N-1$, and $A_{0,N}^{\prime}=\cos\mu\sigma_{Z}+\sin\mu\sigma_{X}$, $A_{1,N}^{\prime}=\cos\mu\sigma_{Z}-\sin\mu\sigma_{X}$.

From the Lemma 1, all partially entangled $N$-partite GHZ states can be self-tested by checking whether the projected state of the remaining two parties ($A_{N-1}$ and $A_{N}$) maximally violates the tilted CHSH inequality. The remaining two parties are the parties after performing local measurements on the other $N-2$ parties. Moreover, for different $r\in\{0,1,\dots,2^{N}-1\}$, the $|GHZ_{\theta}^{r}\rangle$ can all be self-tested by correlations in the Lemma 1 with different measurement settings up to local isometries. In other words, one can obtain a local isometry, such that $\Phi^{r}(|\psi^{r}\rangle)=|junk\rangle|GHZ_{\theta}^{r}\rangle,$ for each $r$. As the isometry can always be constructed by local operations which does not depend on $r$, one can always construct a single isometry, such that $\Phi(|\psi^{r}\rangle)=|junk\rangle|GHZ_{\theta}^{r}\rangle.$ The detailed description will be shown in the next lemma.

Now, let us firstly introduce some notations. For an observable $O^{\prime}$ acting on Hilbert space $\mathcal{H}^{\prime}=\otimes_{i=1}^{N}\mathcal{H}_{A_{i}^{\prime}}$, let $\widetilde{O^{\prime}}^{r}=U^{\prime r{\dagger}}O^{\prime}U^{\prime r}$, where $U^{\prime r}=\otimes_{i=1}^{N}{U^{\prime r}_{A_{i}^{\prime}}}$ acting on $\mathcal{H}^{\prime}$. The $\mathcal{H}_{A_{i}^{\prime}},i\in\{1,2,\dots,N\}$, are two dimensional Hilbert spaces. The unitary operator $U^{\prime r}$ satisfies the equation $U^{\prime r}|GHZ^{r}_{\theta}\rangle=|GHZ^{0}_{\theta}\rangle$ and is constructed by the product of identity matrix $I^{\prime}$, and Pauli matrices $X^{\prime},Z^{\prime}$. Then, one can define $\widetilde{O}^{r}=U^{r{\dagger}}OU^{r}$ by replacing the $I^{\prime},X^{\prime},Z^{\prime}$ in $\widetilde{O^{\prime}}^{r}$ with $I,X,Z$. By the above special unitary transformation, one can obtain following Lemma 2.

From the Lemma 2, self-testing method with fixed measurements can be used to distinguish special entangled pure states. Here, let $\{|GHZ_{\theta}^{r}\rangle\}_{r=0}^{2^{N}-1}$ be reference states and $|GHZ_{\theta}^{0}\rangle$ be a standard reference state. For example, there is a set of states $\{|\psi^{s}\rangle\}_{s=0}^{2^{N}-1}$ shared by $A_{1},A_{2},\dots,A_{N}$. If one shared state $|\psi^{s_{1}}\rangle$ satisfies the correlations in the Lemma 2 with $r=0$, one can specify the shared state $|\psi^{s_{1}}\rangle$ as state $|\psi^{0}\rangle$ according to standard reference state $|GHZ_{\theta}^{0}\rangle$. Then, for another shared state $|\psi^{s_{2}}\rangle$ with $s_{2}\in\{0,1,2,\dots s_{1}-1,s_{1}+1,\dots,2^{N}-1\}$ , if it satisfies correlations in the Lemma 2 for one $r$ with $r\in\{1,2,\dots,2^{N}-1\}$, e.g., $r=3$, then, one resets the $s_{2}$ as $s_{2}=3$. In other words, the state $|\psi^{s_{2}}\rangle$ can be rewritten as $|\psi^{3}\rangle$ and these correlations have self-tested the $|GHZ_{\theta}^{3}\rangle$. Therefore, the states $|\psi^{s_{1}}\rangle$ and $|\psi^{s_{2}}\rangle$ are actually different. Now, the main result of the paper is following.

The detailed proof is shown in Appendix A. Here, we present a brief description. Let the $\tau_{A_{1}A_{2}\dots A_{N}}^{r}=|\psi^{r}\rangle\langle\psi^{r}|$ acting on $\otimes_{i=1}^{N}\mathcal{H}_{i}$ be the state shared by $A_{1},A_{2},\dots,A_{N}$ conditioned on outcome $r$. From the Lemma 2, there exists a single isometry such that $\Phi(|\psi^{r}\rangle)=|junk\rangle|GHZ_{\theta}^{r}\rangle$. By tracing out the subsystems $\mathcal{H}_{1},\dots,\mathcal{H}_{N}$, one can construct a single pair of swap channels $\Gamma_{A_{i}}:\mathcal{L}(\mathcal{H}_{A_{i}})\to\mathcal{L}(\mathcal{H}_{A_{i}^{\prime}}),i\in\{1,2,\dots,N\}$, such that

for all $r$. With the help of Choi-Jamiołkowski map [36], one can construct completely positive and unital maps which are associated with above swap channels, such that

The $2^{N}$ equations given by Eq.  (7) imply that a real measurement $\mathcal{R}=\{R_{R_{1}\dots R_{N}}^{r}\}_{r=0}^{2^{N}-1}$ is capable of simulating ideal tilted GSM, $\{GHZ_{\theta}^{r}\}_{r=0}^{2^{N}-1}$, i.e., the Theorem 1 self-tests the tilted GSM. The method presents a unified form of the theorem for multipartite case without resorting to different Bell inequalities. Furthermore, one can also self-test multipartite GSM, if $\alpha=0,\theta=\pi/4$. Moreover, if $N=3$, one can recover the case of three-qubit GSM [36].

Remarkably, for any self-testing method of tilted GHZ-states, if the ideal measurements in the self-testing procedure are constructed by Pauli matrices, it can be adopted to self-test tilted GSM. Such a property can be a rule to construct the self-testing method for tilted GSM.

The ideal self-testing method is an excellent tool to device-independently certify quantum information tasks. However, due to the imperfection of quantum devices, the accurate correlations in the above theorem may not be satisfied. Hence, a robust version of self-testing is necessary from an experimental point of view. For convenience, we will study here a robust self-testing scheme of three-qubit GSM, where $N=3,\alpha=0,\theta=\pi/4$. The method for studying robustness of other cases is similar.

Before presenting the robustness of GSM, let us firstly study the robust self-testing of the GHZ state with semi-definite programs (SDP) method. One can rewrite $A_{1},A_{2},A_{3}$ as $A,B,C$ and let $A_{i,1}=A_{i},A_{i,2}=B_{i},A_{i,3}=C_{i},i\in\{0,1\}$. Let the state shared by $A,\ B$ and $C$ with outcome $r=0$ be $\tau_{ABC}^{0}=|\psi^{0}\rangle\langle\psi^{0}|$. In a general way, one can adopt the fidelity $F=\langle GHZ|\sigma_{A^{\prime}B^{\prime}C^{\prime}}^{0}|GHZ\rangle$ to capture the distance of the unknown state to the target state [42], where $|GHZ\rangle=\frac{|000\rangle+|111\rangle}{\sqrt{2}}$ and $\sigma_{A^{\prime}B^{\prime}C^{\prime}}^{0}=\Gamma_{A}\otimes\Gamma_{B}\otimes\Gamma_{C}(\tau_{ABC}^{0})$. The maps $\Gamma_{f},f\in\{A,B,C\}$ are defined from Fig.  3 as $\Gamma_{f}(|\xi\rangle_{f}\langle\xi|)=Tr_{\mathcal{H}_{f}}(S_{X_{f},Z_{f}}|0\rangle\langle 0|\otimes|\xi\rangle_{f}\langle\xi|S_{X_{f},Z_{f}}^{{\dagger}})$ with $f\in\{A,B,C\}$. Here, the assumption that $X,Z$ are anti-commutative in the definition of $\Gamma$ has been removed. The state $\sigma_{A^{\prime}B^{\prime}C^{\prime}}^{0}$ can be written as

From the definition of fidelity, one has

where the fidelity can be expressed as a linear function of the expectation values. Suppose the channel suffers with white noise (weight $\epsilon$), one can transform the problem of robustness into the problem that finding a lower bound on the fidelity. It can be solved by SDP [24, 42, 43, 44]:

where $M$ is a moment matrix defined by $M_{ij}=Tr(\tau_{ABC}^{0}D_{i}^{{\dagger}}D_{j})$ with set $\{D_{1}=I,D_{2}=Z_{A},D_{3}=X_{A}\dots\}$ [45]. For an ideal case, the fidelity is 1 when error $\epsilon=0$. For other $\epsilon$ up to 0.1225, the relations between minimal fidelity and error are shown in Fig.  4. Thus, the Fig.  4 gives a lower bound of fidelity for different $\epsilon$. Without loss of generality, one can define the relation between minimal fidelity and $\epsilon$ as a function $G(\epsilon^{0})$, which will be used to study the robustness of GSM. Here, the $\epsilon$ has been rewritten as $\epsilon^{0}$.

For defining quality of real measurement $\mathcal{R}$ as a simulation of ideal GSM $\mathcal{P}$, where $\mathcal{R}=\{R_{R_{1}R_{2}R_{3}}^{r}\}_{0}^{7}$ and $\mathcal{P}=\{GHZ^{r}\}_{0}^{7}$, we directly extend the definition in Ref.  [36] to three parties as

Here, we omit the subscript of $GHZ_{\frac{\pi}{4}}^{r}$ as $GHZ^{r}$ and $\Lambda_{R_{1}},\ \Lambda_{R_{2}},\ \Lambda_{R_{3}}$ are unital CPTP maps with $\Lambda_{R_{1}}:\ \mathcal{L}(\mathcal{H}_{R_{1}})\to\mathcal{L}(\mathcal{H}_{A^{\prime}}),\ \Lambda_{R_{2}}:\ \mathcal{L}(\mathcal{H}_{R_{2}})\to\mathcal{L}(\mathcal{H}_{B^{\prime}}),\ \Lambda_{R_{3}}:\ \mathcal{L}(\mathcal{H}_{R_{3}})\to\mathcal{L}(\mathcal{H}_{C^{\prime}})$. The symbol $\langle\ ,\ \rangle$ for two matrices $L_{1}$ and $L_{2}$ implies

Now, the robust version of self-testing method is presented as following.